No, items of the same type share a cooldown as long as they're on the same hero or companion. Items in different inventories do not share a cooldown. Also note that a Refresher Orb will not refresh its own cooldown even if you have multiple instances of it (in case you might think "hey it refreshes the cooldown of other inventory items first but then they share the reset cooldown").

The most notable instance where the subtle clarification with companions is important is on Lone Druid. You can have items with separate cooldowns on both you and your Spirit Bear. So, a two Hand of Midas build is actually doable and they do not share a cooldown as they are in different inventories. A more common but less intriguing example would be Phase Boots. You can have Phase Boots on both you and your Spirit Bear and they will not share a cooldown either, meaning if you had phased with one recently it will not affect your ability to phase with the other.

Note that this does not apply to things like Beastmaster's hawk and boar as they don't have inventories you can put items in.